"On a splitting scheme for the nonlinear Schrodinger equation in a random medium"

Speaker: 

Professor Renaud Marty

Institution: 

UCI

Time: 

Monday, November 6, 2006 - 11:00am

Location: 

MSTB 256

We consider a nonlinear Schr\"odinger equation (NLS) with random
coefficients, in a regime of separation of scales corresponding to
diffusion approximation. The primary goal is to propose and
study an efficient numerical scheme in this framework. We use a
pseudo-spectral splitting scheme and we establish the order of the
global error. In particular we show that we can take an integration step
larger than the smallest scale of the problem, here the correlation
length of the random medium. We study
the asymptotic behavior of the numerical solution in the diffusion
approximation regime.

Detecting the trail of a random walker in random scenery.

Speaker: 

Professor Noam Berger

Institution: 

UCLA

Time: 

Tuesday, May 9, 2006 - 1:00pm

Location: 

MSTB 254

Let G be a transient graph, and flip a fair coin at each vertex.
This gives a distribution P. Now start a random walk from a vertex v, and
retoss the coin at each visited vertex, this time with probability 0.75
for heads and probability 0.25 for tails. The eventual configuration of
the coins gives a distribution Q. Are P and Q absolutely continuous w.r.t.
each other? are they singular? (i.e. can you tell whether a random walker
had tampered with the coins or not?) In the talk I'll answer to this
question for various graphs and various types of random walk. Based on
joint work with Y. Peres.

Proofs of phase transitions by comparison to mean-field theory

Speaker: 

Professor Marek Biskup

Institution: 

UCLA

Time: 

Tuesday, June 6, 2006 - 11:00am

Location: 

MSTB 254

Mean-field theory is one of the most standard tools used by
physicists to analyze phase transitions in realistic systems. However,
regarding rigorous proofs, the link to mean-field theory has been
limited to asymptotic statements which do not yield enough control
of the actual systems. In this talk I will describe a new approach to
this set of problems -- developed jointly with Lincoln Chayes and
Nicolas Crawford -- that overcomes this hurdle in a rather elegant
way. As a conclusion, I will show that a general, ferromagnetic
nearest neighbor spin system on Z^d undergoes a first order phase
transition whenever the mean-field theory indicates one, provided
the dimension d is sufficiently large. Extensions to systems with non
nearest neighbor interactions will also be discussed.

On phase transitions for homopolymers and hetereopolymers.

Speaker: 

Professor Michael Cranston

Institution: 

UCI

Time: 

Tuesday, January 31, 2006 - 11:00pm

Location: 

MSTB 254

Given the measure on random walk paths $P_0$ and a Hamiltonian $H$ the Gibbs perturbation of $H$ defined by
$$\frac{dP_{\beta,t}}{dP_0}=Z^{-1}_{\beta,t}\exp\{\beta H(x)\}$$
with
$$Z_{\beta,t}=\int \exp\{-\beta H(x)\}dP_0(x)$$
gives a new measure on paths $x$ which can be viewed as polymers.
In the case $H(x)=\int_0^t\delta_0(x_{s})ds(\int_0^t\delta_0(x_{s})dW_s)$ we say the resulting measure is concentrated on "homopolymers" ("heteropolymers") and are interested in the influence of dimension and $\beta$ on their behavior.

Phase coexistence of gradient Gibbs measures

Speaker: 

Professor Marek Biskup

Institution: 

UCLA

Time: 

Tuesday, February 21, 2006 - 11:00pm

Location: 

MSTB 254

A gradient Gibbs measure is the projection to the gradient variables $\eta_b=\phi_y-\phi_x$
of the Gibbs measure of the form
$$
P(\textd\phi)=Z^{-1}\exp\Bigl\{-\beta\sum_{\langle x,y\rangle}V(\phi_y-\phi_x)\Bigr\}\textd\phi,
$$
where $V$ is a potential, $\beta$ is the inverse temperature and $\textd\phi$ is the product
Lebesgue measure. The simplest example is the (lattice) Gaussian free field
$V(\eta)=\frac12\kappa\eta^2$. A well known result of Funaki and Spohn (and Sheffield)
asserts that, for any uniformly-convex $V$, the possible infinite-volume measures of this type are
characterized by the \emph{tilt}, which is a vector $u\in\R^d$ such that
$E(\eta_b)=u\cdot b$ for any (oriented) edge $b$. I will discuss a simple example
for which this result fails once $V$ is sufficiently non-convex thus showing that
the conditions of Funaki-Spohn's theory are generally optimal. The underlying
mechanism is an order-disorder phase transition known, e.g., from the context
of the $q$-state Potts model with sufficiently large $q$. Based on joint work
with Roman Koteck\'y.

The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus.

Speaker: 

Professor Jason Schweinsberg

Institution: 

UCSD

Time: 

Tuesday, February 14, 2006 - 11:00pm

Location: 

MSTB 254

Let x and y be points chosen uniformly at random
from the four-dimensional discrete torus with side length n.
We show that the length of the loop-erased random walk from
x to y is of order n^2 (log n)^{1/6}, resolving a conjecture
of Benjamini and Kozma. We also show that the scaling limit
of the uniform spanning tree on the four-dimensional discrete
torus is the Brownian continuum random tree of Aldous. Our
proofs use the techniques developed by Peres and Revelle,
who studied the scaling limits of the uniform spanning tree
on a large class of finite graphs that includes the
d-dimensional discrete torus for d >= 5, in combination with
results of Lawler concerning intersections of
four-dimensional random walks.

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